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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Long-time behavior for a nonlinear fourth-order parabolic equation

Authors: María J. Cáceres, J. A. Carrillo and G. Toscani
Journal: Trans. Amer. Math. Soc. 357 (2005), 1161-1175
MSC (2000): Primary 35K65, 35B40, 35K35
Published electronically: August 11, 2004
MathSciNet review: 2110436
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Abstract: We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.

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Additional Information

María J. Cáceres
Affiliation: Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

J. A. Carrillo
Affiliation: ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 - Bellaterra, Spain

G. Toscani
Affiliation: Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy

Keywords: Asymptotic behavior, entropy dissipation, degenerate parabolic equation, diffusion equation
Received by editor(s): November 13, 2002
Received by editor(s) in revised form: October 2, 2003
Published electronically: August 11, 2004
Additional Notes: The authors were partially supported by the European IHP network “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications”, RNT2 2001 349 and Spanish-Italian bilateral HI01-175. The first and second authors acknowledge support from DGI-MCYT/FEDER project BFM2002-01710. The third author acknowledges partial support from the Italian Minister for Research, project “Mathematical Problems in Kinetic Theories”.
Article copyright: © Copyright 2004 American Mathematical Society